I can't find any post talking about that, so, just wondering if anything is in progress ? I plan to try to implement it through a plugin maybe, but could be also directly in PhpBB or at least, in Prosilver

Nope. And it would probably not be something we would pursue as workbox is, generally speaking, for websites that provide mostly static content, which a forum is not. Note how in the benefits they don't mention site data.

Offline support is for so called web applications often used in Progressive Web Apps. The idea is for you to "install" the web application via a shortcut and after you load it the data that it's supposed to display will be downloaded locally as storage for the site and used if for some reason the device isn't online or the site is unreachable. This can happen, for instance, if the device is a tablet that doesn't have a 4G card or if the site is an intranet application only accessible through VPN or while on the company's wifi. This only works with small subsets of data though. I can't imagine the entirety of what is displayed for a forum to be downloaded locally. Browsers wouldn't allow that much data to be downloaded for a site, at least not easily.

There are some web applications where this makes sense (think about a company phone directory, or a schedule of events, or even a kitchen menu for a work location), but certainly doesn't sound very useful for a forum that has thousands of posts, topics, and users where the data would be constantly changing.